I can’t help with the east coast as we went in Feb so focussed on inland areas and south / west coast, but it’s a really easy and cheap country to travel around.
We did 2.5 weeks for about £1k each including flights - no kids, just me and my husband. We stayed in apartments or small hotels, basic but clean, and didn’t hire a driver, so saved a lot there and actually much preferred the train journeys. They can be long but if you reserve seats in advance they aren’t crowded and are actually pretty comfortable - and the views make them so worthwhile!
This website has all you need for train travel: www.seat61.com/SriLanka.htm
As for places to go, Sigiriya is a must see. We stayed in the village for 4 nights which made a really good base for us. Go to Sigiriya rock, visit Dambulla caves (easy tuktuk journey), or hire a driver for the day to go on an elephant safari in one of the national parks nearby (we did Minneriya) and visit Polonnaruwa (hire a bike and cycle round the ancient temples).
Kandy is a bigger city round a lake, nice for a night or 2, the temple of the tooth is worth a look, and the British Garrison cemetery (colonial graves - find the caretaker, he’ll tell you the interesting stories!)
Nuwara Eliya is very British, lots of nice hiking around at Horton Plains, Ella is beautiful and hilly with the tea plantations around it.
I’ve not been, but have heard good things about Arugam Bay on the east coast.